Expert: Paul Klarreich - 7/27/2009

Question
Let the sum of 'a sub n' from n=1 to infinity be a real series that converges. Which of the following is true? (Please explain)

(a) The series: Sum of (a sub n)squared converges

(b) The series: Sum of (a sub n)Cubed converges

(c) The series: Sum of (a sub n)to the one-third power converges

Answer
Questioner: Jason
Country: United States
Category: Calculus
Private: No
Subject: Convergence
Question: Let the sum of 'a sub n' from n=1 to infinity be a real series that converges. Which of the following is true? (Please explain)

(a) The series: Sum of (a sub n)squared converges

(b) The series: Sum of (a sub n)Cubed converges

(c) The series: Sum of (a sub n)to the one-third power converges
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Hi, Jason,

If  SUM a[n] converges, then lim a[n] = 0 (required for convergence of a series.)

That means there is some N such that  for all n > N, a[n] < 1.

In that case, for all n > N,  a[n]^2 (and a[n]^3, for that matter) is less than  a[n].

Then SUM a[n]^2 and SUM a[n]^3  converge by the comparison test.
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But  SUM a[n]^1/3 might not converge.

Ex:  Let  a[n] = 1/n^3.

Now this is a p-series and converges.

But  a[n]^1/3 is 1/n. And  SUM 1/n is the harmonic series which diverges.